Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Matching:

Unconditional) biased estimates
ofbs
Conditional )unbiased estimates
ofbs

V. Overview on Statistical
Inferences for Logistic
Regression

Chap. 5: Statistical Inferences
Using Maximum Likelihood Tech-
niques

Statistical inferences involve the
following:
 Testing hypotheses

 Obtaining confidence intervals

Quantities required from computer
output:

1. Maximized likelihood valueLð^uÞ


2. Estimated variance–covariance
   matrix
   covariances off
   the diagonal

variances on
diagonal

V(q) =

Note:covdð^y 1 ;^y 2 Þ¼r 12 s 1 s 2

If we consider the other parameters in the

model for matched data, that is, the bs,

the unconditional likelihood approach gives

biased estimates of thebs, whereas the condi-

tional approach gives unbiased estimates of

thebs.

We have completed our description of the

ML method in general, distinguished between

unconditional and conditional approaches, and

distinguished between their corresponding like-

lihood functions. We now provide a brief over-

view of how statistical inferences are carried

out for the logistic model. A detailed discussion

of statistical inferences is given in the next

chapter.

Once the ML estimates have been obtained, the

next step is to use these estimates to make

statistical inferencesconcerning the exposure–

disease relationships under study. This step

includes testing hypotheses and obtaining con-

fidence intervals for parameters in the model.

Inference-making can be accomplished through

the use of two quantities that are part of the

output provided by standard ML estimation

programs.

The first of these quantities is themaximized

likelihood value, which is simply the numerical

value of the likelihood function L when

the ML estimates (u^) are substituted for their

corresponding parameter values (y). This value

is calledL(u^) in our earlier notation.

The second quantity is theestimated variance–

covariance matrix. This matrix,V^of^u, has as its

diagonal the estimated variances of each of the

ML estimates. The values off the diagonal are

the covariances of pairs of ML estimates. The

reader may recall that the covariance between

two estimates is the correlation times the stan-

dard error of each estimate.

Presentation: V. Overview on Statistical Inferences for Logistic Regression 117